$f'(x)=\dfrac{3f(x)}{x\ln(x)}$ Is $f(x)=2(\ln(x))^3$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: In order to find whether $f(x)=2(\ln(x))^3$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $f(x)$, we need to find the corresponding $f'(x)$ expression to substitute into the equation: $\begin{aligned} f'(x)&=\dfrac{d}{dx}\left[2(\ln(x))^3\right] \\\\ &=\dfrac{6(\ln(x))^2}{x} \end{aligned}$ Now we substitute ${f(x)=2(\ln(x))^3}$ and ${f'(x)=\dfrac{6(\ln(x))^2}{x}}$ into the equation: $\begin{aligned} {f'(x)}&=\dfrac{3{f(x)}}{x\ln(x)} \\\\ {\dfrac{6(\ln(x))^2}{x}}&\stackrel{?}{=}\dfrac{3\left({2(\ln(x))^3}\right)}{x\ln(x)} \\\\ \dfrac{6(\ln(x))^2}{x}&\stackrel{?}{=}\dfrac{6(\ln(x))^3}{x\ln(x)} \\\\ \dfrac{6(\ln(x))^2}{x}&\stackrel{\checkmark}{=}\dfrac{6(\ln(x))^2}{x} \end{aligned}$ We obtained the same expression on each side. In conclusion, yes, $f(x)=2(\ln(x))^3$ is a solution to the differential equation.